What is the maximum number of dichotomies in a square?
The number of dichotomies of 4 data points will clearly be $2^4 = 16$. According to these slides the definition of dichotomy in context of Statistical Learning is:



  Different ‘hypotheses’ over the finite set of $N$ input points.


Which basically means hypotheses with unique behaviours over the input points. Two or more different hypotheses can have same behaviour on the data points (consider the case of a square covering the $4$ data points, an even larger square will also cover the $4$ data points. Thus they are different hypotheses but have same behaviour) and hence emphasis on the term unique.


The proof of axis aligned squares having $\mathcal V \mathcal C$ dimension $3$ can be found here. It's pretty straightforward so I don't want to explain it here.